In this work we extend the application of the implicit function theorem, where gradients of inner optimization result with respect to parameters are found as in many existing works, [1]}, [2]}, [3]}, and derive second-order derivatives, i.e., the IFT Hessian. We leverage this result for three main contributions: (i) We show that the computational complexity of obtaining second-order derivatives is, in many cases, still dominated by the same matrix inversion bottleneck required for the IFT gradient and so our method can be implemented equally efficiently. (ii) We analyze our IFT Hessian expression to derive computational complexity and error bound expressions. We derive a new form of the regularized error bound under diagonal regularization of the matrix inverse operation in the application of IFT. (iii) We use our second-order derivative expression to apply second-order optimization methods to two machine learning datasets and show that these methods lead to faster bilevel optimization, requiring fewer lower level problem evaluations.
